
where \(\hat{p}_{i, \alpha }, \hat{x}_{i, \alpha }, m_{i, \alpha }\) , and \(\omega _{i, \alpha }\)  are the momentum, position, mass, and frequency of the \(\alpha \) th oscillator of the \(i\) th bath, respectively.
The system operator that represents the effects of the \(i\) th local heat bath is given by \(\hat{V}_{i}\) , and \(k_{i, \alpha }\)  is the coupling constant for the interaction between the system and the \(\alpha \) th oscillator of the \(i\) th bath.
The second term in \(\hat{H}_{I}\)  is a counterterm that compensates for the renormalization of the potential energy caused by the first term in \(\hat{H}_{I}.\)[1]}
Depending on the situation, we may choose a local or nonlocal system–bath interaction by the way in which we set \(\hat{V}_{i}\) : for high-frequency intramolecular vibrational modes, we consider the case in which each LO oscillator is coupled to its own bath, which is expressed as \(\hat{V}_{i} = \hat{b}_{i}^{\dagger } + \hat{b}_{i}\) , whereas for low-frequency intermolecular vibrational modes or phonon modes, we consider the case in which multiple LO oscillators are coupled to a single global bath, described as \(\hat{V}_i \rightarrow \hat{V}_{0} = \hat{b}^{\dagger }_{0} + \hat{b}_{0}\) , with \(i\)  being ignored in the summations in Eqs. (REF ) and ().
If necessary, we can further treat a nonlocal heat bath described by \(\hat{V}_{ij}\)  between the \(i\) th and \(j\) th sites, which often plays an important role in determining the efficiency of an energy transfer process.[2]}, [3]}
